The function f is defined as follows.
f left parenthesis x right parenthesis equalsleft brace Start 2 By 1 Matrix 1st Row 1st Column 2 x 2nd Row 1st Column 1 EndMatrix Start 2 By 1 Matrix 1st Row 1st Column if x not equals 0 2nd Row 1st Column if x equals 0 EndMatrix
 
 
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​(b) Locate any intercepts.
​(c) Graph the function.
​(d) Based on the​ graph, find the range.
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Part 1
​(a) The domain of the function f is 
  
enter your response here.
​(Type your answer in interval​ notation.)

Let's solve each part of the problem one more time for clarity.

### (a) Domain of the function
The function \( f(x) \) is defined as:
\[ 
f(x) = 
\begin{cases} 
2x & \text{if } x \neq 0 \\ 
1 & \text{if } x = 0 
\end{cases}
\]

The function \( f(x) \) is defined for all real numbers \( x \), so the domain of \( f(x) \) is:
\[ 
(-\infty, \infty)
\]

### (b) Intercepts
#### x-intercept
The x-intercept is where the function crosses the x-axis, which means \( f(x) = 0 \).
- For \( x \neq 0 \), \( 2x = 0 \) implies \( x = 0 \). However, at \( x = 0 \), \( f(0) = 1 \), so there is no x-intercept.

#### y-intercept
The y-intercept is where the function crosses the y-axis, which is at \( x = 0 \).
- For \( x = 0 \), \( f(0) = 1 \).
So, the y-intercept is at \( (0, 1) \).

### (c) Graph of the function
To graph the function \( f(x) \), we note the following:
- For \( x \neq 0 \), \( f(x) = 2x \). This is a line with a slope of 2 passing through the origin, but we need to exclude the point at \( x = 0 \).
- For \( x = 0 \), \( f(x) = 1 \). This is a point at \( (0, 1) \).

Here is the graphical representation of \( f(x) \):

1. For \( x \neq 0 \), the graph is the line \( y = 2x \) with an open circle at \( (0, 0) \) indicating that the point is not included.
2. At \( x = 0 \), there is a filled circle at \( (0, 1) \).

I will now plot the graph for you.

### (d) Range of the function
The range of a function is the set of all possible output values (y-values) of the function.

- For \( x \neq 0 \), \( y = 2x \) can take any real value.
- At \( x = 0 \), \( y = 1 \).

Thus, the range of the function \( f(x) \) includes all real numbers except for 1 because the function value 1 is explicitly defined only for \( x = 0 \). Hence, the range of the function is:
\[ 
(-\infty, \infty)
\]

### Summary
(a) The domain of \( f \) is \( (-\infty, \infty) \).

(b) The function has no x-intercept and a y-intercept at \( (0, 1) \).

(c) The graph consists of the line \( y = 2x \) for \( x \neq 0 \) and a point at \( (0, 1) \).

(d) The range of \( f \) is \( (-\infty, \infty) \).

Now, let's plot the graph for you.Here is the graph of the function \( f(x) \):

1. The blue line represents \( y = 2x \) for \( x \neq 0 \).
2. The red point represents \( (0, 1) \).
3. The open circle at the origin indicates that \( (0, 0) \) is not included in the function.

### Summary
(a) The domain of \( f \) is \( (-\infty, \infty) \).

(b) The function has no x-intercept and a y-intercept at \( (0, 1) \).

(c) The graph consists of the line \( y = 2x \) for \( x \neq 0 \) and a point at \( (0, 1) \).

(d) The range of \( f \) is \( (-\infty, \infty) \).

Do you want more details or have any questions about these solutions? Here are 5 related questions you might want to explore:

1. How do you find the domain of a more complex piecewise function?
2. Can you determine the range of a piecewise function with more than two pieces?
3. How do you graph a piecewise function with different types of functions (e.g., polynomial and exponential)?
4. How do intercepts of a function affect its graph?
5. What is the importance of continuity in piecewise functions?

**Tip:** When graphing piecewise functions, it’s helpful to use different markers (e.g., open or closed circles) to clearly indicate where the function is defined and where it is not.

Explore the function f(x) = { 2x if x ≠ 0, 1 if x = 0 }. Learn about its domain, range, intercepts, and graph representation with detailed explanations and visual aids. Suitable for high school students studying algebra and functions.

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